On the measure-theoretic entropy and topological pressure of free semigroup actions

Abstract

We compute the exact value of Voiculescu's perturbation theoretic entropy of the boundary actions of free groups. This result is a partial answer of Voiculescu's question.

A variational principle for free semigroup actions

“…The formula (5.1) also generalizes the statement of Theorem 1.1 of [20]: when a = p, it simplifies to…”

“…Yet, we are looking for intrinsic dynamical and ergodic concepts, the less dependent on the skew product the better. In [20], the authors proposed a notion of measure theoretical entropy for free semigroup actions and probability measures invariant by all the generators of the semigroup, but only a partial variational principle was obtained there (as happened in [5] and [10]). The major obstruction to use this strategy, in order to extend the ergodic theory known for a single expanding map, is the fact that probability measures which are invariant by all the generators of a free semigroup action may fail to exist.…”

Quantitative recurrence for free semigroup actions

“…The formula of Abramov and Rokhlin [2] for the measure theoretical entropy of the skew product F G with respect to the product measure P × ν suggests a way to define a fibred notion of metric entropy of a free semigroup action with respect to a random walk on Σ + p and an invariant measure on X, which we will denote by h ν (S, P). This will be done in Subsection 5.1, just before proving a partial variational principle which extends Theorem 1.2 of [16] to non-symmetric random walks. Meanwhile, recall that P (q) top (F G , 0, P) stands for the quenched topological pressure of the skew product F G with respect to the random walk P (see [5]), h top (S) is the topological entropy of the free semigroup action S (cf.…”

“…[21,Theorem 28]). Moreover, if P × ν has positive entropy with respect to F G then, using (16), for P × ν-almost every (ω, x), one has (cf. [3,25])…”

“…Moreover, we shall study the connection between the maximum hitting frequency/fastest mean return time to a set with its size when estimated by different invariant measures, extending the ergodic optimization obtained in [14] to the random context we are considering. Finally, we will introduce the notion of measure-theoretic entropy of a semigroup action and obtain a partial variational principle which improves the estimate in [16] and complements [6,7]. We refer the reader to Subsection 2.2 for the precise statements of the main results.…”